# How to interpret Einstein Notation across equals sign?

My apologies if this has been asked before. I wasn't able to find an explanation that made sense to me on Google.

Suppose we have the equation below, which is written in Einstein notation:

$$y_{j} = x_iz_ix_j$$

and say that both $j$ and $i$ take on values of $1$ and $2$. Is this notation to be interpreted as

$$y_1 = (x_1z_1+x_2z_2)x_1$$ $$y_2 = (x_1z_1+x_2z_2)x_2$$

or as $$y_1+y_2 = (x_1z_1+x_2z_2)(x_1+x_2)?$$

In other words, does the index across the equals sign imply multiple equations, or adding up terms on each side of a single equation?

Context: I am an undergrad math major studying fluid flow. The conservation of momentum equation was written in Einstein Notation and I am having trouble understanding the meaning.

• It's the first. Think about the simpler case $x_i = y_i$. This should mean that the vectors $x$ and $y$ are equal, not that their components sum to the same number. – knzhou Feb 21 '17 at 0:28

The correct interpretation is the first one: $$y_j=x_iz_ix_j$$ Means that to obtain the jth component of $y$ you have to sum over the repeated indeces, i.e.
$$y_j=\sum_{i=1}^2 x_iz_ix_j$$